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Properties

Label 10.2.635...504.1
Degree $10$
Signature $[2, 4]$
Discriminant $6.358\times 10^{19}$
Root discriminant \(95.57\)
Ramified primes $2,3,41$
Class number $5$ (GRH)
Class group [5] (GRH)
Galois group $D_{10}$ (as 10T3)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972)
 
Copy content gp:K = bnfinit(y^10 + 28*y^8 - 82*y^7 - 408*y^6 - 3280*y^5 - 13117*y^4 - 11480*y^3 - 7798*y^2 - 2624*y - 5972, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972)
 

\( x^{10} + 28x^{8} - 82x^{7} - 408x^{6} - 3280x^{5} - 13117x^{4} - 11480x^{3} - 7798x^{2} - 2624x - 5972 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $10$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 4]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(63580957267604373504\) \(\medspace = 2^{15}\cdot 3^{5}\cdot 41^{8}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(95.57\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{1/2}41^{4/5}\approx 95.57244712596375$
Ramified primes:   \(2\), \(3\), \(41\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{6}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{10}a^{8}+\frac{1}{10}a^{7}+\frac{1}{10}a^{6}+\frac{1}{10}a^{5}-\frac{1}{2}a^{4}-\frac{3}{10}a^{3}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{45\cdots 50}a^{9}+\frac{58\cdots 76}{22\cdots 75}a^{8}+\frac{27\cdots 57}{45\cdots 50}a^{7}+\frac{32\cdots 57}{45\cdots 50}a^{6}+\frac{21\cdots 31}{45\cdots 50}a^{5}+\frac{10\cdots 07}{45\cdots 50}a^{4}-\frac{42\cdots 89}{22\cdots 75}a^{3}+\frac{13\cdots 39}{45\cdots 50}a^{2}+\frac{19\cdots 78}{45\cdots 15}a-\frac{44\cdots 07}{22\cdots 75}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{5}$, which has order $5$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{10}$, which has order $10$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $5$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{82337378}{140434539175}a^{9}-\frac{1279569}{140434539175}a^{8}+\frac{2219789121}{140434539175}a^{7}-\frac{6634764279}{140434539175}a^{6}-\frac{36255657457}{140434539175}a^{5}-\frac{259131564829}{140434539175}a^{4}-\frac{1058428211609}{140434539175}a^{3}-\frac{684862066358}{140434539175}a^{2}+\frac{32024120768}{28086907835}a-\frac{673606233367}{140434539175}$, $\frac{3921704044467}{90\cdots 30}a^{9}+\frac{82385235832957}{90\cdots 30}a^{8}-\frac{151223715500134}{45\cdots 15}a^{7}+\frac{814131272728896}{45\cdots 15}a^{6}-\frac{12\cdots 83}{90\cdots 23}a^{5}+\frac{43\cdots 52}{45\cdots 15}a^{4}-\frac{32\cdots 61}{18\cdots 46}a^{3}-\frac{39\cdots 97}{90\cdots 30}a^{2}-\frac{30\cdots 54}{45\cdots 15}a-\frac{19\cdots 37}{90\cdots 23}$, $\frac{33\cdots 37}{90\cdots 30}a^{9}-\frac{394640613816131}{90\cdots 30}a^{8}+\frac{45\cdots 02}{45\cdots 15}a^{7}-\frac{14\cdots 73}{45\cdots 15}a^{6}-\frac{68\cdots 29}{45\cdots 15}a^{5}-\frac{53\cdots 63}{45\cdots 15}a^{4}-\frac{41\cdots 91}{90\cdots 30}a^{3}-\frac{26\cdots 87}{90\cdots 30}a^{2}+\frac{72\cdots 34}{90\cdots 23}a+\frac{14\cdots 06}{45\cdots 15}$, $\frac{334920560229003}{64\cdots 50}a^{9}-\frac{989726826909989}{64\cdots 50}a^{8}+\frac{12\cdots 01}{64\cdots 50}a^{7}-\frac{62\cdots 99}{64\cdots 50}a^{6}+\frac{42\cdots 23}{64\cdots 50}a^{5}-\frac{12\cdots 79}{64\cdots 50}a^{4}-\frac{56\cdots 87}{32\cdots 25}a^{3}-\frac{48\cdots 04}{32\cdots 25}a^{2}-\frac{43\cdots 44}{64\cdots 45}a-\frac{36\cdots 31}{32\cdots 25}$, $\frac{70\cdots 83}{64\cdots 50}a^{9}+\frac{768176723131281}{64\cdots 50}a^{8}+\frac{19\cdots 71}{64\cdots 50}a^{7}-\frac{55\cdots 29}{64\cdots 50}a^{6}-\frac{30\cdots 37}{64\cdots 50}a^{5}-\frac{23\cdots 19}{64\cdots 50}a^{4}-\frac{46\cdots 72}{32\cdots 25}a^{3}-\frac{39\cdots 44}{32\cdots 25}a^{2}-\frac{28\cdots 04}{12\cdots 89}a+\frac{20\cdots 89}{32\cdots 25}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 582192.1229160837 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 582192.1229160837 \cdot 5}{2\cdot\sqrt{63580957267604373504}}\cr\approx \mathstrut & 1.13794760834986 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 + 28*x^8 - 82*x^7 - 408*x^6 - 3280*x^5 - 13117*x^4 - 11480*x^3 - 7798*x^2 - 2624*x - 5972); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{10}$ (as 10T3):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{6}) \), 5.1.180848704.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: 20.0.4042538127064933401151627374468337238016.3
Degree 10 sibling: 10.0.7947619658450546688.1
Minimal sibling: 10.0.7947619658450546688.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{5}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.2.0.1}{2} }^{5}$ ${\href{/padicField/17.2.0.1}{2} }^{5}$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.2.0.1}{2} }^{5}$ R ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{5}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.1.2.3a1.2$x^{2} + 10$$2$$1$$3$$C_2$$$[3]$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
\(3\) Copy content Toggle raw display 3.5.2.5a1.2$x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(41\) Copy content Toggle raw display 41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)